# Photon entanglement: why three angles?

1. Aug 29, 2014

### Jabbu

When the two polarizers are set 60 degrees apart, for example, QM prediction is 25% correlation. It is already different than what is believed to be classical or "expected" result. So what is the point of testing more than one angle in a single experiment? And what difference does it make when those angles are shuffled randomly instead of tested separately?

2. Aug 29, 2014

### DrChinese

The issue is whether local realism is compatible with the predictions of Quantum Mechanics.

Local realists (such as you as best I can tell) believe there are particle attributes even when not observed. If so, there must be counterfactual values. A third reading (even if not taken) would be part and parcel of that belief. Otherwise, there is no point is asserting local realism. So that explains the 3, as 2 are predicted by QM (but a third is not).

Testing with fast switching is not strictly needed, but shows that there is no possibility of communication between the observing devices by an unknown mechanism at light speeds or less. Since that experiment was performed already (Aspect, Weihs, etc) we know that is not an issue. You have previously been given a reference on Weihs et al.

Combine the above with Bell's Theorem, and we know that local realism is not viable.

Now, please do not ask the same question over again. You know already where that leads. If you don't understand my answer, explain what you don't understand. I believe you already have a reference to my web page on Bell, but if not:

http://www.drchinese.com/Bells_Theorem.htm

3. Aug 29, 2014

### Staff: Mentor

Two angles, not perpendicular to one another, are enough to validate the quantum mechanical prediction fot the correlation at two angles. However, Bell's theorem makes a stronger statement: No local hidden variable theory (loosely speaking, one in which the unmeasured properties of the particle have definite values) can reprodiuce the predictions of quantum mechanics. It takes three angles to test this propostion because with two partciles we get two measurements; we need a third angle to have an unmmeasured angle in every set of measurements we make.

Have you read Bell's paper yet? http://www.drchinese.com/David/Bell_Compact.pdf

4. Aug 30, 2014

### Jabbu

Can you describe "must be counterfactual values" in terms of photons, polarizers and data recorded?

One theta already confirms cos^2(theta), it's already non-local. Can you explain in more practical terms what objection or explanation local realists have so more is needed to convince them?

Do you have some page about "DrC challenge"? From what I've seen in other posts about it, I don't understand why the "data sample" required is in this format:

a: + + - + - - +
b: - - + - + + -
c: - + - + + - +

...shouldn't it be:

aAlice: + + - + - - +
aBob: - - + - + + -

bAlice: + + - + - - +
bBob: - - + - + + -

cAlice: + + - + - - +
cBob: - - + - + + -

...where a, b, c are three theta relative angles between Alice and Bob polarizers? Can you show how do you calculate results?

5. Aug 30, 2014

### Staff: Mentor

When I measure the polarization of one member of the pair on one angle and I measure the polarization of the other member of the pair on another angle, then I know the polarization of both members of the pair on those angles. The polarization on any other angle is "counterfactual" - I didn't measure it.

We can construct a local realistic theory for the $cos^2\theta$ result easily enough: just say that the photons are created in a state of definite polarization in every direction, with the values of the polarization in each direction such that the $cos^2\theta$ law holds on average across many pairs . This theory is said to be "counterfactually definite" because it asserts that the polarizations I didn't measure still have definite values that I would have obtained if I had measured them.

The first column of this example above should be read as "for the first pair if we measure the polarization on angle A the left-hand photon will pass and the right-hand one will not; if we measure on angles B or C the left-hand photon will not pass and the right-hand one will". The second column should be read as "for the second pair if we measure the polarization on angle B the left-hand photon will not pass and the right-hand one will; if we measure on angles A or C the left-hand one will pass and the right-hand one will not". This is exactly the local realistic theory I describe above - both photons are created with definite polarization values at all three angles.

You calculate the results by choosing any two of the three possible results because we only get to make two measurements, one on each photon. The challenge is to construct a data set that will lead to a violation of Bell's equality no matter which measurements we choose to make on each pair - and if you try it you'll find that it cannot be done. Therefore, no theory in which the results of measurements on all three angles are predetermined can match the experimental results.

6. Aug 30, 2014

### DrChinese

The DrChinese Challenge, yes very good Jabbu! You've been doing some study!! Keep reading the references, especially as to how they formulate their setups and theoretical predictions - Weihs, Dehlinger, Zeilinger, etc. Now what I say below is a bit of a different take than Nugatory's but they are fundamentally the same.

Let's talk about a SINGLE photon we will call Alice. Realists believe it has polarization properties at all times. So there must be values for any 3 angles a, b, c (0/120/240 are what I use, which as you have said is the same as 0/60/-60). Further, as pointed out by EPR (1935), for entangled photon pairs, realism implies that they are actually *predetermined*. That is because the outcome of ANY measurement on Alice can be predicted with certainty in advance (by measuring the SAME property a, b or c on entangled partner Bob). This conclusion was very reasonable (until Bell came along). A realist simply believed there was a more complete specification of Alice than QM allows. But otherwise there was no specific contradiction, it's simply a matter of your interpretation. (Back then: Bohr v. Einstein.)

1. But it turned out that the requirement that Alice had simultaneous predetermined polarizations at a, b and c (to be consistent with the above paragraph) led to severe constraints that were missed early on after EPR. Specifically: it should matter not which of a, b and c you measure on Alice, since the outcome is predetermined. In fact, there should be a data set of values that would match up to any measurement you can do on Alice on some set of runs (let's say you are measuring a) where you can also hypothesize b and c. Creating such dataset is easy if we stop here - all you do is provide ANY answer for the unmeasured angles b and c on Alice and no one could disprove it. We'll start with:

a: + + - + - - + +
b: - - + - + + - +
c: - + - + + - + -

2. Of course, Bob's polarizations must be predetermined as well. So by measuring entangled Bob at one of the other angles (let's say b), you actually learn some additional information* about Alice (at least, that's what realists believe). So you could update your dataset above so that it at least kept the attribute that a and b match the cos^2(theta) predictions of QM. That means there would be no disagreement with QM (which presumably also makes experimentally correct predictions). So your dataset has a good relationship between Alice@a and Bob@b (which tells you Alice@b too). So we revise the dataset to fix this:

a: + + - + - - + +
b: - - + - - + - +
c: - + - + + - + -

a-b is 25% (since cos^2(120) is 25% and note that Type I vs Type II is not an issue here)

3. And it shouldn't matter which of the other 2 angles (b or c) you measure for Bob either, for similar reasons (as both Alice AND Bob's polarizations are all predetermined). So now you update your dataset so Alice@a and Bob@b AND Alice@a and Bob@c both match the QM predictions. This is getting progressively harder to do, but you can do it. You will have a 25% match ratio for each in our example. What we are actually doing is describing Alice by measuring one attribute on Alice and inferring another value by measuring Bob. But we are still talking about Alice at the base. Here is our revised sample dataset.

a: + + - + - - + +
b: - - + - - + - +
c: - + + - - + - -

a-b is 25% (since cos^2(120) is 25%)
a-c is 25% (since cos^2(120) is 25%)

4. But now something funny happens. We fixed it so the match ratio for Alice's a and b is 25% (theta=120 degrees), and the match ratio for Alice's a and c is 25%(theta=120 degrees), BUT... the match ratio for Alice's b and c is now 75% (for same theta=120 degrees). What happened? This is an inconsistent result. Because we specified above that all the angle settings were predetermined, so it shouldn't matter that we are examining Alice's a-b, b-c, or a-c. We will try our best to fix this:

a: + + - + - - + +
b: - - + - - + - +
c: - + + + + + - -

a-b is 25% (since cos^2(120) is 25%)
a-c is 25% (since cos^2(120) is 25%)
b-c is 50% (oops - can't get a combo to be 25%)

So for this to work out, there must be something special about the pair we choose to measure - but that violates Observer Independence (i.e. the results were predetermined). The only way to make this work out is to assume there is some force or information moving from Alice's measuring apparatus to Bob's (or vice versa). And if we fixed it so the selection of the angle pair occurred AFTER Alice and Bob separate, we could determine if such signal occurs at light speeds or faster.

And this is the DrChinese challenge, to come up with values for a, b and c for a set of 8 runs that match what you would see if Alice and Bob were entangled and had predetermined outcomes independent of the measurements performed. This time we had a loser, can you do better?

Of course, experiments show that if there is such an effect, it must be at least 10,000 times faster than c. The other thing is that we can drop the assumption (constraints) of realism we started with in 1. above and that resolves things.

*In fact, such information exceeds the bounds of the HUP! Because you could simply measure a complementary (non-commuting value on Bob and now you would know both about Alice).

Last edited: Aug 30, 2014
7. Aug 30, 2014

### Jabbu

Predetermined at what point in time? Flip of my coin may be predetermined at the beginning of time, but that doesn't mean it can be predicted with more than 50% certainty.

Non-localists (QM) think measurement on Alice can be predicted with certainty in advance? And local realists (classical physics) think outcome can be predicted by Malus law probability?

8. Aug 30, 2014

### RUTA

Here is an example from Mermin using just two particles and two settings. He acknowledges the idea is originally due to Hardy.

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9. Aug 30, 2014

### Jabbu

Here is example with only one setting: theta= 0 degrees.

QM predicts cos^2(0) = 100% correlation.

Classical physics (Malus law) does not predict anything unless photons polarization relative to their prospective polarizer angle is known at the time of photon-polarizer interaction, or known to be uniformly random.

If it is assumed photon pairs come out unpolarized (uniformly random) then classical physics predicts 50% chance for +/- detection on both sides regardless of any polarizer angle. That is 50% chance AB recorded pairs will match (++ or --), and 50% chance they will be mismatch (+- or -+). Match - mismatch = 0% correlation.

Completely opposite predictions for just one angle setting, how is this not sufficient? The only way Malus law can predict 100% correlation for theta= 0 is if it is known photon pairs come out polarized at 0 degrees as well. Therefore, if you can prove photons come out with uniformly random polarization relative to their polarizer you have proved non-locality.

10. Aug 31, 2014

### Staff: Mentor

One angle setting is sufficient to verify the quantum mechanical $cos^2\theta$ prediction for that angle and to falsify the classical prediction.

The three-angle experiments you've been reading about are looking (generally successfully) for violations of Bell's inequality for three observables. These lead to an even stronger conclusion: no local hidden-variable theory (not just the classical theory that we've already falsified) can be correct.

11. Aug 31, 2014

### RUTA

If both photons' polarizations are totally random with respect to each other, you would find four possible outcomes ++, +-, -+, -- for your single setting distributed evenly which means 50% correlation, i.e., ++ and -- each occurring 25% of the time. Once you find you only get ++ and -- outcomes each 50% of the time, you have entanglement. You can explain this via Mermin's "instruction sets," i.e., local hidden variables. You need another setting to establish counterfactuals and violate Bell's inequality. Read the paper.

12. Aug 31, 2014

### stevendaryl

Staff Emeritus
No, that just proves that Malus' law isn't the correct description.

13. Aug 31, 2014

### stevendaryl

Staff Emeritus
Hardy came up with a thought experiment that doesn't involve any inequality.

Prepare two electrons in the composite state $\frac{1}{\sqrt{3}}(|U_z\rangle |U_z\rangle + |U_z\rangle |D_z\rangle + |D_z\rangle |U_z\rangle)$

where $|U_z\rangle$ means spin-up in the z-direction, and $|D_z\rangle$ means spin-down in the z-direction.

Use this state for an EPR-like experiment involving Alice and Bob. Alice can perform one of two experiments: measure spin in the x-direction, or the z-direction. For each direction, she gets two possible results: spin-up or spin-down. Similarly for Bob.

We can prove, using quantum mechanics:

1. If Alice measures spin-up in the z-direction, then Bob will measure spin-up in the x-direction.
2. If Bob measures spin-up in the z-direction, then Alice will measure spin-up in the x-direction.
3. If Alice measures spin-down in the z-direction, then Bob will measure spin-up in the z-direction.

Let's try to explain this result under the assumption that the outcomes are predetermined. Then there are five possible values for the "hidden variable":

$\lambda_1$: Both Alice and Bob measure spin-up in either direction.
$\lambda_2$: Alice measures spin-up in either direction. Bob measures spin-down in z-direction, or spin-up in x-direction.
$\lambda_3$: Alice measures spin-up in z-direction or spin-down in x-direction. Bob measures spin-down in z-direction, or spin-up in x-direction.
$\lambda_4$: Alice measures spin-down in z-direction or spin-up in x-direction. Bob measures spin-up in z-direction, or spin-down in x-direction.
$\lambda_5$: Alice measures spin-down in z-direction or spin-up in x-direction. bob measures spin-up in either direction.

Something that can never happen, according to a hidden-variables theory, is this:
Alice and Bob both measure spin-down in the x-direction. The proof is this:
• Alice either has spin-up in the z-direction, or spin-down in the z-direction.
• If it is spin-up, then Bob must have spin-up in the x-direction. (Fact #1)
• If it is not spin-up in the z-direction, then Bob must have spin-up in the z-direction. (Fact #3)
• But if Bob has spin-up in the z-direction, then Alice will have spin-up in the x-direction. (Fact #2)

So the assumption that the results are predetermined implies that it is impossible for both Bob and Alice to measure spin-down in the x-direction.

Quantum-mechanically, though, it is possible. (I think that happens 1/12 of the time)

14. Aug 31, 2014

### Jabbu

Malus law is the only classical and locally causal solution. Proving that it is not correct (complete) description is the whole point, because every other description is no longer classical or locally causal.

What do you believe is classical prediction for theta= 0 degrees?

Last edited: Aug 31, 2014
15. Aug 31, 2014

### Jabbu

If photons polarization is uniformly random with respect to polarizers, then for Malus law it is irrelevant whether relative polarization between photons in each pair is constant (entangled) or random as well, because the chance for each photon to mark either "+" or "-" stays 50% in either case.

I believe in a previous thread we established mismatches are counted as well and that correlation = matches - mismatches, not just number of matches. It's not terribly important since 50% prediction is still far from 100%. Do you have some reference for this "correlation" formula?

Exactly. It's like you and me are tossing two coins and every time it's a match. Can a result be any more "non-local" than that? Your coin flips should not be correlated with my coin flips, and if they are, you can call it "entanglement", but more practically said it's a pretty crazy display of non-local causality. No other relative angle makes this more clear. 100% correlation is the most unambiguously bizarre correlation there can possibly be, testing other angles is almost pointless compared to theta = 0. Is this not true?

Last edited: Aug 31, 2014
16. Aug 31, 2014

### stevendaryl

Staff Emeritus
Malus' law is certainly not the only possible local theory. It's not even the only local theory that is consistent with classical E&M in the limit that quantum effects become negligible.

17. Aug 31, 2014

### stevendaryl

Staff Emeritus
Why do you think that the only possibilities are Malus' law, or nonlocal interactions?

18. Aug 31, 2014

### stevendaryl

Staff Emeritus
That isn't necessarily nonlocal. Suppose that the coins worked this way:

If the nth digit of $\pi$ is an even number, then the nth coin flip will be heads. Otherwise, the nth flip will be tails.​

That's a perfectly local rule, in the sense that the two coins don't need instantaneous communication to achieve it. They just need a little microprocessor and some internal mechanism for shifting weight to one side or the other.

Last edited: Aug 31, 2014
19. Aug 31, 2014

### Jabbu

If you mean to suggest there is some other locally causal (classical physics) equation which can calculate probability for photon-polarizer interaction outcome, you only need to name it.

20. Aug 31, 2014

### stevendaryl

Staff Emeritus
If you are claiming that something is impossible, then it's up to you to produce a proof. Otherwise, you're just guessing. You asked "why three angles"? Because with three angles we can prove that the results of QM cannot be reproduced by a local realistic model. With a single angle, there is no such proof. It's as simple as that.

"I can't think of a way to do it" does not logically imply "There is no way to do it".